Reliability of signal transmission in stochastic nerve axon equations

TL;DR

This paper develops a statistical method to estimate the probabilities of spontaneous activity and propagation failure in stochastic nerve axon models, using membrane potential integrals and linearization techniques.

Contribution

It introduces a novel estimator based on membrane potential integrals and provides an analytical approximation for failure probabilities in conductance-based neuronal models.

Findings

The integral of membrane potential effectively detects activity and failure.

Analytical expressions approximate probabilities using linear stochastic differential equations.

Numerical results validate the approximation for Hodgkin-Huxley models.

Abstract

We introduce a method for computing probabilities for spontaneous activity and propagation fail- ure of the action potential in spatially extended, conductance-based neuronal models subject to channel noise, based on statistical properties of the membrane potential. We compare different estimators with respect to the quality of detection, computational costs and robustness and propose the integral of the membrane potential along the axon as an appropriate estimator to detect both spontaneous activity and propagation failure. Performing a model reduction we achieve a simplified analytical expression based on the linearization at the resting potential (resp. the traveling action potential). This allows to approximate the probabilities for spontaneous activity and propagation failure in terms of (classical) hitting probabilities of one-dimensional linear stochastic differential equations. The quality of the approximation with respect to the noise amplitude is discussed and illustrated with numerical results for the spatially extended Hodgkin-Huxley.